If you want Galaxian, you would likely need to rewire the entire cabinet. The only way around this would be if you could find an adapter to convert Phoenix to Galaxian, which may not exist. The only thing I know of is an adapter to play Phoenix in a JAMMA-wired cabinet, which is the exact opposite of what you're looking for.
Or, you could simply *make* an adapter... You know, the pre-made adapters are cool and all, but it's so easy to make them yourself. And the bonus is, you can make an adapter to go from anything to anything else.
Galaxian is very, very similar to Pac-Man. The pinout is almost exactly the same, and it runs off AC in the original cabinet. But, it will run happily off DC just like Pac will.
If you've not done a lot of soldering, an adapter is a very, very simple project that should help you hone your skills. The pads are large and easy to work with, and there are no components, only wire. Just print out the pinouts of Phoenix, and Galaxian, and match up the signals. Here's a good starter guide, he's building a Pac to Jamma adapter, but the construction/principle is the same.
http://www.therealbobroberts.net/adaptor.html
Bob even sells an adapter "kit", for ten bucks, which contains an edge connector, fingerboard, and wire. The only thing you'll have to do is cut down the "jamma" side of the fingerboard to mate with the Phoenix harness. This process is described here:
http://www.therealbobroberts.net/p2j.html
Maybe if you ask him nicely, he'll even cut it down for you.
Here's the pinout for Phoenix. It uses an 18/36 edge connector:
http://www.mikesarcade.com/cgi-bin/spies.cgi?action=url&type=pinout&page=Phoenix.txt
And here's the pinout for Galaxian. It uses a 22/44 edge connector:
http://www.mikesarcade.com/cgi-bin/spies.cgi?action=url&type=pinout&page=Pac.html
Note that this chart is Pac-Man, but the pins that are different on Galaxian are noted to the side.
So, all you do, is get the fingerboard, cut one end down to 18 pins wide, solder a 22/44 edge connector on the other end, and connect up the signals.
Then, just plug in Galaxian and go. And if you ever want to swap back to Phoenix, it'll be simple to unplug the adapter and swap it back. Phoenix is a good game, but hard when you first start playing it. Give it a little time...
-Ian